RATIO. 



verginff fraftions will be — , — , 7—7 ^, , 



^*' a a6 + 1 (ab + l) c + a 



(6c -r l) d -r i . , . , .„ , ^ 



; — ^^ — ; ; , &c. which will be found 



l{ab + i)c + a^.a + ab + 1 



to agree with the preceding rule ; and thefe fraftions will 

 be altL-rnately greater and lefs than that propofed, which will 

 be the lall of the feries. 



Let u3 illuftrate this rule by an example. Required a 



feries of converging fraftions towards - ■, which is 



° 1 00000 



the fradlion commonly employed for exprelfing the ratio of 

 the diameter to the circumference of a circle. 



Operation by Divi/ion. 



100000)314159(3 = a 



300000 



to denote the ratio of the diameter to the circumference 

 of a circle, and amongft our approximations we find fome 

 of tliofe commonly employed by former writers ; as 7 to 

 22, being that given by Archimedes ; 106 to 3^^, which 

 is another ratio formerly ufcd ; alfo 115 10355, being that 

 invented by Peter Melius. Each of t'liefc ratios is alter- 

 nately too fmall and too great, to exprefs the ratio of the 

 diameter to the circumference of the circle. Thus, i to i 

 is too fmall a ratio, 7 to 22 too great, 106 to ^j^ too fmall, 

 and 1 13 to 355 too great; and fo on. 



We may therefore divide the above feries into two didinft 

 clafles, one of which fliall exhibit the ratios all in exccfs, 

 and the other all in defed, which will ftand as follows -. 

 viz. 



Ratios in defeS. 

 3^ 333 9208 76149 

 I 106' 2931' 24239' 

 Ratios in excefs. 



2J 355 95^ 3'4'59 

 7' 113' 3044* looooo" 

 And between every two confecutive fraftions we may now 

 interpolate as many fraftions of the fame kind, as is one 

 lefs than the number of units in the quotients from which 

 it was formed. If we write the firft of the above feries, 

 with the correfponding quotients above their terms, we 

 (hall have 



15 



333^ 

 106' 



25 

 9208 



2931' 



7 



79H9 



Having thus obtained our quotients, the ieveral fraftions 

 will be eafily formed by the preceding rule ; thus, 



3 7 15 I 25 I 7 4 



1 ^ M3 355 9^ 95^ 76149 JH'jQ. 

 i' 7' 106' 113' 2931' 3044' 2423' looooo' 



the laft of which is the fame as the original fraftion. Tliefe 

 fraftions are formed according to the preceding rule, which 

 will be underftood from one example ; thus, the fifth frac- 

 tion is formed from the two preceding fraftions as follows- : 



355 X 25 -I- 333 = 9208 numerator, 



113 X 25 4- 106 = 29^1 denominator, 



and all the others are obtained in the fame manner. 



Each of thefe frattions reprefents an approximate ratio 

 towards the original one, and each of them nearer than 

 any preceding fraftioii in the ieries, and nearer than any 

 fraftion expreffed in lefs numbers, 01 than any fraftion 

 having a lets denominator than the lucceeding fraftions. 



Thus, - is a nearer approximation than any fraftion whofe 



denominator is lefs than 3, and - - nearer than anv frac- 



113 



tion whofe denominator is lefs than 293 1 ; &c. 



Our original fraftion exhibited the ratio ufually employed 



24239 



We may therefore interpolate, between the two firft 

 fraftions, fourteen others, which will poflefs the fame pro- 

 perty as the principal fraftions above, -vi-z,. of approxi- 

 mating nearer to the true ratio than any other fraftions 

 expreffed in lefs terms, but all in defeft. Between the 

 fecond and third we may interpolate twenty-four fraftions : 

 and between the third and fourth, fix : all of them lefs 

 than the propofed ratio. But if we take the fecond fe- 

 ries and their correiponding quotients, they will itand 

 thus ; 



III 4 



22 355 95^ 3'4i59 

 7' 113' 3044' looooo' 

 which (hews, that no fraftion can here be interpolated 

 either between the firft and fecond, or between the fecond 

 and third ; but there may be three interpolated between 

 the third and fourth, which will, of courfe, be all in ex- 

 cefs. This interpolation is performed as follows : to the 

 numerator and denominator of the lefs fraftion add, once, 

 twice, three times, &c. the numerator and denominator of 

 the principal fraftion which interpofes between the two, 

 which vsfill form the interpolated fraftions required. Thus, 

 becaufe^^'^9 



24239 



is the principal fraftion interpofed between 



^-^ and , the intermediate or interpolated frac- 



3044 I 00000 



tious will be 



9563 4-76149 



3044 

 9563 



-!- 24239 

 + 2.76149 



_ 85712 



27283 



_ 161861 



3044 4- 2.24239 "" 51522 



9563 + 3-76149 _ 238010 

 3044 + 3.24239 75761 

 3 M 2 



And 



